# Urcohomologies and cohomologies of $N$-complexes

### Naoya Hiramatsu

Okayama University, Japan### Goro C. Kato

California Polytechnic State University, San Luis Obispo, USA

## Abstract

For a general sequence of objects and morphisms, we construct two $N$-complexes. Then we can define cohomologies $(i,k)$-type of the $N$-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of $N$-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of $N$-complexes are given.

## Cite this article

Naoya Hiramatsu, Goro C. Kato, Urcohomologies and cohomologies of $N$-complexes. Port. Math. 67 (2010), no. 4, pp. 511–524

DOI 10.4171/PM/1875